Cross-correlation of instantaneous phase increments in pressure-flow fluctuations: applications to cerebral autoregulation

Abstract

We investigate the relationship between the blood flow velocities (BFV) in the middle cerebral arteries and beat-to-beat blood pressure (BP) recorded from a finger in healthy and post-stroke subjects during the quasisteady state after perturbation for four different physiologic conditions: supine rest, head-up tilt, hyperventilation, and CO2 rebreathing in upright position. To evaluate whether instantaneous BP changes in the steady state are coupled with instantaneous changes in the BFV, we compare dynamical patterns in the instantaneous phases of these signals, obtained from the Hilbert transform, as a function of time. We find that in post-stroke subjects the instantaneous phase increments of BP and BFV exhibit well-pronounced patterns that remain stable in time for all four physiologic conditions, while in healthy subjects these patterns are different, less pronounced, and more variable. We propose an approach based on the cross-correlation of the instantaneous phase increments to quantify the coupling between BP and BFV signals. We find that the maximum correlation strength is different for the two groups and for the different conditions. For healthy subjects the amplitude of the cross-correlation between the instantaneous phase increments of BP and BFV is small and attenuates within 3-5 heartbeats. In contrast, for post-stroke subjects, this amplitude is significantly larger and cross-correlations persist up to 20 heartbeats. Further, we show that the instantaneous phase increments of BP and BFV are cross-correlated even within a single heartbeat cycle. We compare the results of our approach with three complementary methods: direct BP-BFV cross-correlation, transfer function analysis, and phase synchronization analysis. Our findings provide insight into the mechanism of cerebral vascular control in healthy subjects, suggesting that this control mechanism may involve rapid adjustments (within a heartbeat) of the cerebral vessels, so that BFV remains steady in response to changes in peripheral BP.

FIG. 1

BP and BFV signals during CO₂ rebreathing after a bandpass Fourier filter in the range [0.05 Hz, 10 Hz] and normalization to unit standard deviation: (a), (b) for a healthy subject, (c), (d) for a post-stroke subject.

FIG. 2

Cross-correlation function C(τ) for the BP and BFV signals during four physiologic conditions: (a) for the same healthy subject and (b) for the same post-stroke subject as shown in Fig. 1. BP and BFV signals are preprocessed using a bandpass Fourier filter in the range [0.05 Hz, 10 Hz] and are normalized to unit standard deviation before the analysis. Since BFV precedes BP, the maximum value C_max in the cross-correlation function C(τ) is located not at zero lag but at τ≈−0.1 sec.

FIG. 3

(Color online) (a) The relative phases φ_BFV(t)–φ_BP(t) for healthy subjects (a) and post-stroke subjects (b), where φ(t) is the instantaneous phase of a signal. The phase difference of BFV and BP for both groups may fluctuate around a constant value or jump between different constant values. These jumps are always a multiple of 2π and occur due to (1) artifacts in the BP signals related to machine calibration during the process of recording which do not occur in the simultaneously recorded BFV signals or (2) difference in the morphology of the BP and BFV waves (see Fig. 1) which on certain occasions exhibit a more strongerly pronounced bump on the right shoulder of the wave for one of these two signals. The distributions of Ψ_1,1≡(2π)⁻¹[φ_BFV(t)–φ_BP(t)] mod 1 for healthy subjects and for post-stroke subjects are shown in (c) and (d), respectively. Employing this definition for Ψ_1,1 eliminates the effect which jumps of multiple of 2π may have on the synchronization analysis. The number of bins in the histogram is N=40.

FIG. 4

(Color online) (a) The relative phases φ_BFV(t₁)–φ_BP(t₂) for a healthy subject and a post-stroke subject. We find that the relative phases between BFV and BP depend on the time difference t₂–t₁ between these two signals. Accordingly, the distributions of Ψ_1,1≡(2π)⁻¹[φ_BFV(t₁)–φ_BP(t₂)] mod 1, shown in (b) and (c) for a healthy subject and a post-stroke subject, respectively, also depend on the time difference between BFV and BP. We note that our choice of time difference t₂–t₁=0.4 sec is arbitrary and does not carry any specific physiologic meaning. The number of bins in the histogram is N=40.

FIG. 5

Presentation of the BP and BFV signals vs their Hilbert transforms (a), (b) and their corresponding instantaneous phase increment Δφ during the CO₂ rebreathing condition (c), (d) for the same data from a healthy subject as shown in Figs. 1(a) and 1(b). BP and BFV signals vs their Hilbert transforms (e), (f) and their corresponding instantaneous phase increment Δφ during the CO₂ rebreathing condition (g), (h) for the same data from a post-stroke subject as shown in Figs. 1(c) and 1(d). Repetitive temporal patterns associated with each heartbeat in Δφ for the peripheral BP signal from a healthy subject (c) are not matched by corresponding patterns in the cerebral BFV signal (d), reflecting active cerebral vascular regulation. In contrast, periodic patterns in Δφ of the peripheral BP signal from a post-stroke subject (g) are matched by practically identical patterns in Δφ of the cerebral BFV signal (h), indicating dramatic impairment of cerebral vascular tone with higher vascular resistance after minor ischemic stroke.

FIG. 6

Cross-correlation function C(τ) of the instantaneous phase increment Δφ for the BP and BFV signals during four physiologic conditions. We find that the cross-correlation function for all healthy subjects exhibits a very distinct type of behavior compared to post-stroke subjects. Two typical examples are shown. Left: a healthy subject: C(τ) has a small amplitude at τ=0 and is close to zero at time lags τ<5 seconds during all four conditions. Right: a post-stroke subject: C(τ) has a much larger amplitude at τ=0 which lasts for lags τ up to 20 seconds, indicating a strong coupling between the BP and BFV signals—i.e., loss of cerebral autoregulation.